低秩Hankel矩阵去噪方法研究
On Low-Rank Hankel Matrix Denoising
摘要：
线性系统中的低复杂性假设通常可以表示为具有广义Hankel结构的数据矩阵的秩亏。这使得有可能通过估计潜在的结构化低秩矩阵来对数据进行降噪。然而，标准的低秩近似方法不能保证在估计无噪声矩阵时表现良好。本文综述了近年来奇异值收缩法在矩阵去噪中的研究成果。针对低秩Hankel矩阵去噪问题，提出了一种基于数据驱动奇异值收缩的结构化低秩近似迭代算法.数值结果表明，在输入输出轨迹去噪和脉冲响应去噪问题中，该方法在估计无噪声矩阵方面优于现有的低秩矩阵逼近和去噪算法。
英文摘要：
The low-complexity assumption in linear systems can often be expressed as rank deficiency in data matrices with generalized Hankel structure. This makes it possible to denoise the data by estimating the underlying structured low-rank matrix. However, standard low-rank approximation approaches are not guaranteed to perform well in estimating the noise-free matrix. In this paper, recent results in matrix denoising by singular value shrinkage are reviewed. A novel approach is proposed to solve the low-rank Hankel matrix denoising problem by using an iterative algorithm in structured low-rank approximation modified with data-driven singular value shrinkage. It is shown numerically in both the input-output trajectory denoising and the impulse response denoising problems, that the proposed method performs the best in terms of estimating the noise-free matrix among existing algorithms of low-rank matrix approximation and denoising.
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